Tiers, Idle Tiers

The distinctive tiered seeding system used in professional tennis avoids the phenomenon of seeding waves – local effects in the seeding order in which it may be beneficial to be seeded lower rather than higher. This is a good thing.

But it does it at the expense of creating another seeding anomaly – the equal treatment of players with a given tier. Thus at the Western and Southern, where the tiers are 1, 2, 3-4, 5-8, 9-12, and 13-16, it makes no difference to a player’s prospects whether he is seeded 3 rather than 4, 5 rather than 6, 7, or 8, and so forth.

But it does matter, and matter a good deal, when a player crosses a tier boundary, being seeded 2 rather than 3, 8 rather than 9, and so forth. In this post, I’ll try to put a dollar figure on the gain or loss associated with each of these boundaries.

First, note that 1 and 2 are not really distinct tiers – they have exactly the same equity (apart from the difference in skill they presumably represent). But being known as the world number one is richly rewarded, no doubt, in terms of sponsorship opportunities. I won’t try to quantify those, though I suspect that they dwarf the figures I’ll associate with crossing other tier boundaries.

For this experiment, I’ll use the expectations associated with a given position in a perfectly seeded tourney with those in which the seeding is imperfect only at the tier boundaries. Thus for the base case, the seeds are in strict order of skill. In the other cases, seeds are in skill except that the individual players, without changing skill levels, are swapped across one of the tier boundaries. In the first run, the second best player is demoted one level and the third best promoted one level, so that the former is in the 3-4 tier and the latter in the 2 tier. And so forth for the other runs, swapping the 4 and 5 seeds, the 8 and 9 seeds, the 12 and 13 seeds, and the 16 seed with the best of the otherwise unseeded players.

The results show that some of the tier boundaries are more significant than others.

expectation

seed

change

percentage

$ 373,608

2, demoted

$ (17,091)

-4.57%

$ 284,022

3, promoted

$ 15,149

5.33%

$ 244,552

4, demoted

$ (26,489)

-10.83%

$ 191,433

5, promoted

$ 23,492

12.27%

$ 148,780

8, demoted

$ (41,066)

-27.60%

$ 99,485

9, promoted

$ 39,226

39.43%

$ 83,392

12, demoted

$ (6,217)

-7.06%

$ 73,631

13, promoted

$ 5,457

6.55%

$ 65,160

16, demoted

$ (8,177)

-11.62%

$ 55,194

17, promoted

$ 7,551

11.13%

The effect of swapping the 2 and 3 seeds is modest, at least in terms of the percentage of change in expectation. Here, as in other cases, it’s not just the two swapped seed that are affected. The 1 seed loses nearly $6000 of equity because the 2/3 swap increases the chance that it will encounter the true second-best player in the semifinal rather than the final. Other high seeds benefit a little from the disorder at the top.

The most dramatic tier boundary is the one between the 5-8 tier and the 9-12 tier. That’s because, at the Western and Southern, the 5-8 tier players get a bye, and the 9-12 do not. And the boundary between getting a seed at all and not getting one is also a bit larger than the one that precedes it.

In each case, the small perturbation in the relationship between seeds and skill is reflected in a small negative effect on the fairness (C) statistic. Fairness (C) for the base case is 17.772. For the other cases, in order of the seeds swapped, the figures are 18.009, 17.897, 17.818, 17.777, and 17.792.